Preprint A689/2011
Homogenization of Degenerate Porous Medium Type Equations in Ergodic Algebras
Jean Carlos Silva | Frid, Hermano
Keywords: homogenization | porous medium equation | algebras with mean values | ergodic algebras | two-scale Young measures
We consider the homogenization problem for general porous medium type equations of the form $u_t=\D f(x,\frac{x}{\ve}, u)$. The flux function $f(x,y,\cdot)$ may be of two different types. In the type~1 case, $f(x,y,\cdot)$ is strictly increasing; this is a mildly degenerate case. In the type~2 case, $f(x,y,\cdot)$ has the form $h(x,y)F(u)+S(x,y)$, where $F(u)$ is just a nondecreasing function; this is a strongly degenerate case. We address both, the Cauchy problem and the initial-boundary value problem, with null boundary condition. The homogenization is carried out in the general context of ergodic algebras. As far as the authors know, homegenization of such degenerate quasilinear parabolic equations is addressed here for the first time. We also review the existence and stability theory for such equations and establish new results needed for the homogenization analysis. Further, we include some new results on algebras with mean value, specially a new criterion establishing the null measure of level sets of elements of the algebra, which is useful in connection with the homogenization of porous medium type equations in the type~2 case.